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%I #8 Mar 15 2021 13:32:54
%S 1,1,6,63,790,10896,159783,2445499,38627339,625074945,10310825610,
%T 172747624244,2931749347058,50298296409267,870968173547912,
%U 15202903269996956,267229314929729201,4726210689606805342,84045293131256519705,1501874060469604610959
%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(5*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(5*n+4)).
%C Equals row k = 5 of rectangular table A340940.
%F Given g.f. A(x), the following sums are all equal:
%F (1) B(x) = Sum_{n>=0} x^n*A(x)^(5*n)/(1 - x*A(x)^n),
%F (2) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^(5*n+1)),
%F (3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(5*n+4)),
%F (4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+5)),
%F (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+5))),
%F (6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(5*n^2+5*n) * (1 - x^2*A(x)^(10*n+5)) / ((1 - x*A(x)^(5*n+1))*(1 - x*A(x)^(5*n+4)));
%F see the example section for the value of B(x).
%e G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 790*x^4 + 10896*x^5 + 159783*x^6 + 2445499*x^7 + 38627339*x^8 + 625074945*x^9 + 10310825610*x^10 + ...
%e such that the following sums are all equal:
%e B(x) = 1/(1-x) + x*A(x)^5/(1 - x*A(x)) + x^2*A(x)^10/(1 - x*A(x)^2) + x^3*A(x)^15/(1 - x*A(x)^3) + x^4*A(x)^20/(1 - x*A(x)^4) + ...
%e and
%e B(x) = 1/(1-x*A(x)) + x*A(x)^4/(1 - x*A(x)^6) + x^2*A(x)^8/(1 - x*A(x)^11) + x^3*A(x)^12/(1 - x*A(x)^16) + x^4*A(x)^16/(1 - x*A(x)^21) + ...
%e also
%e B(x) = 1/(1-x*A(x)^4) + x*A(x)/(1 - x*A(x)^9) + x^2*A(x)^2/(1 - x*A(x)^14) + x^3*A(x)^3/(1 - x*A(x)^19) + x^4*A(x)^4/(1 - x*A(x)^24) + ...
%e further
%e B(x) = 1/(1-x*A(x)^5) + x/(1 - x*A(x)^6) + x^2/(1 - x*A(x)^7) + x^3/(1 - x*A(x)^8) + x^4/(1 - x*A(x)^9) + x^5/(1 - x*A(x)^10) + ...
%e where
%e B(x) = 1 + 2*x + 8*x^2 + 60*x^3 + 640*x^4 + 8085*x^5 + 112116*x^6 + 1649968*x^7 + 25311223*x^8 + 400396030*x^9 + 6485530349*x^10 + ...
%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
%o H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+5)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(5*m+4)) ), #A)/4; A=H); W=A;A[n+1] }
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A340940, A340941, A340942, A340894, A340895, A340943.
%Y Cf. A341958.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 16 2021
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